Reduced Model Reconstruction Method for Stable Positive Network Systems

We consider a reconstruction problem of a reduced stable positive network system with the preservation of the original interconnection structure based on an
$H^{2}$
optimal model reduction problem with constraints. To this end, we define an important set using the Perron–Frobenius theory of nonnegative matrices such that all elements of the set are stable and Metzler. Using the projection onto the set, we propose a cyclic projected gradient method to produce a better reduced model than an initial reduced model in the sense of the
$H^{2}$
norm. In the method, we use Lipschitz constants of the gradients of our objective function to define the step sizes without a line search method whose computational complexity is large. Moreover, the existence of the Lipschitz constants guarantees the global convergence of our proposed algorithm to a stationary point. The numerical experiments demonstrate that the proposed algorithm improves a given reduced model, and can be used for large-scale systems.